Those are called anti-afterfire solenoids. Ford-six is right, most people don't know how to idle the engine down when they are done mowing.
I can tell you though that they are fairly essential on twin cylinder engines (I know yours is a single cyl).
Wjat happens is simple: when you shut off an engine, you are really just killing the spark. As the engine continues to spin without spark, it cycles unburnt fuel through the cylinder and pushes it into the hot exhaust. Sometimes you'll be 4-5 steps away from the tractor when the fuel vapors in the exhaust explode. It'll make you jump every time.
Those solenoids are little more than a spring loaded electro magnet (ie door bell ringer) yet they can be pretty expensive. I replace one on a Kohler powered JD and it was about $40. I recently replaced one on a Kawasaki twin powered JD and it was closer to $140!! They really have you by the short hairs because there are not yet many (if any) after market suppliers for these things.
I don't have access to my B&S prices on this computer and the Briggs technicians site is down for maintenance this morning, so I can't give you a part number or price for that solenoid, but I will get it to you.
I suggest you try replacing that first. They go bad all the time. The alcohol in fuel is commonly blamed, but sediment/water can also be the cause. They really can't usually be fixed, but you can test it by the following procedure:
Shut off the fuel to the carb with the shut off valve or carefully pinch the fuel line closed with a pair of vice-grips.
Unplug the hot wire connector from the solenoid.
Unscrew the solenoid from the bottom of the carburetor. Gas will run out of the bowl, so have a rag handy.
Reconnect the hot wire to the solenoid.
Turn the ignition key to the run position.
You should see the plunger snap backwards into the shell of the solenoid. If it doesn't move at all or moves slowly, remove the black ground wire from the block and be sure the metal connector and block contact point are clean and reattach and try again. If it still doesn't move, replace the solenoid.
When you turn the key to the run position, the solenoid is energized and the plunger should snap back into the solenoid, allowing fuel to enter the main jet of the carburetor. When you turn the key to the off or stop position, the spring should snap the plunger upward against the jet to stop any further fuel from flowing into the jet.
You can replace it with a simple bowl screw, but you will have to idle the engine down at shut down, let it run at idle for 1 minute or so, then shut it off. It shouldn't backfire doing it this way, but it still might, then you either have to just get use to it (and reconcile yourself with a shorter muffler life) or take the plunge (no pun intended) and buy a new solenoid.
I'll get you a part number and MSRP this afternoon.
Any local shop should be able to get you one and you can save $ on shipping.

 

Replacement solenoid is B&S pn 692734 and lists for about $38.

Bowl nut is pn 691657 and lists for about $3

Your engine was built in 1995. No surprise if your solenoid is shot by now.
I have had some go bad within a year.

 

Thanks for the great help !
I am still dealing with the flu,but getting better today .
I'll keep you posted .

JD

 

Time to resurrect this thread from last October.
I got sick with fungal pneumonia ,in the hospital for 5 months ! I lost 60 lbs.
I am now too weak to work on the engine. A local shop is picking it up tomorrow.
When I started it yesterday,it was back firing while idling.

 

Jeeze-Louise! I am glad to hear that you are OK.
That is a heck of a way to lose some weight! Next time, just try some portion control or counting calories!

I'd be thrilled to hear what the shop determined to be the problem.
Let us know.

 

Thanks for the response.
I'll be sure to let you know .

 

They dropped off the Murray today.
The tech cleaned the gas cap vent and carb. bowl,seems to be running fine.

 

Blastomycosis, by any chance? Just had a co-worker come down with that a month ago. Was very nasty. He's up and around now but months away from full recovery.

Glad to hear you are on the mend now.